 Correct bit-by-bit

Write neatly, give yourself plenty of space, always write each line below the previous line (leaving lots of blank space on the right of the page if necessary), and do big, clear diagrams with a ruler.

Try not to have your working run over from one page to the next. If a question is going to need long working, then start a new page with it. Be concise in your working (this isn’t GCSE: you don’t have to show every tiny step).

Then, if an answer looks wrong, or your working shows something different from the formula you are asked to prove, don’t cross out everything and start again.

Probably the problem is a small slip

Just go through your working line by line. Find the missing minus, or the misreading from the question, or whatever has caused the problem, then make your correction neatly and follow it forward through your working.

Matrices

To check an inversion A−1, check A A−1 = I

To check a multiplication AB, check det (AB) = det(A).det(B)

If you have a Casio 991ES-Plus or similar calculator, you can check matrix multiplications and inversions on the calculator

To check when you are asked to work out the inverse of a matrix with elements including a variable x, put x=0 or x=1 or some other easy-to-work-with number, and check your answer works out right

Complex numbers

If you’re asked something like (p+2i)/(3+pi), put p=0 or p=1 or some other easy-to-work-with number, and check your answer works out right

Use Pol(x,y) on your calculator to calculate arg(x+iy) and |x+iy|. Use the Alpha key to get your answers as exact surds or fractions of π: click here to see how. Check by drawing a diagram of the complex number and marking in the angle arg(z). Second check by doing tan−1(y/x), remembering this may give you an answer out by π.

To find the remaining roots of a cubic or a quartic, you can use the clunky long division method as a check on the sum-of-roots/product-of-roots method, or the sum-of-roots/product-of-roots method as a check on the clunky long division method. Also, substitute your answers back into the original equation and see they give the evaluation 0.

If you have a Casio 991ES-Plus or similar calculator, you can check roots of cubics or quartics by calculating them on the calculator

Numerical methods

Substitute back your answer into the original equation f(x)=0. You should get an answer near to zero, but not exactly zero.

With linear interpolation, do a good, precise diagram, and check your answer against the diagram. The diagram should show, for example, which of the two previous guesses the new guess is closer to.

Series and induction

If you have a Casio 991ES-Plus or similar calculator, you can check “sum from 10 to 50”-type problems directly on the calculator.

If you’re asked to find a formula for sum of a series, check your answer by plugging in n=1 and n=2 and seeing your formula gives the right sum for those easy cases.

Parametrics

Your best check here is a good, big, clear diagram. Does your answer (where the tangents meet, or whatever), look right on the diagram?

Remember for the parabola all negative t (all past time) is in the bottom of the parabola, t=0 is at the origin, the point where the parabola “turns round”, and all positive t (the future) is in the top of the parabola

For the hyperbola, all negative t is in the bottom left-hand quadrant (from big negative t way out on the left, near the x axis, to small negative t near the y axis). The equation fails for t=0. All positive t is in the top right-hand quadrant (from small positive t near the y axis to big positive t way out on the right, near the x axis)